Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3e 7-1
Chapter 7Chapter 7 Currency Options & Options Markets Currency Options & Options Markets7.1 What is an Option?7.2 Option Payoff Profiles7.3 Profit and Loss on Currency Options7.4 At-the-Money Options7.5 The Determinants of Currency Option Values7.6 Combinations of Options7.7 Hedging with Currency Options7.8 Exchange Rate Volatility Revisited (Advanced)7.9 Summary
Appendix 7-A Currency Option Valuation
Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3e 7-2
A forward obligationA forward obligation A £1 million obligation due in four months
Underlying transaction
Currency exposure
-£1,000,000
V$/£
S$/£
Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3e 7-3
A forward hedgeA forward hedge Buy £1 million in the forward market at
the forward price F1$/£ = $1.45/£
Long pound forward
Exposure of forward contract
+£1,000,000V$/£
S$/£-$1,450,000
Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3e 7-4
An option hedgeAn option hedge A currency option is like one-half of a
forward contract- the option holder gains if pound sterling rises- the option holder does not lose if pound
sterling falls
Long pound call(option to buypound sterling)
S$/£
V$/£
Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3e 7-5
CME pound Dec 1450 call CME pound Dec 1450 call (American)(American)
Type of option: a call option to buy pounds
Underlying asset: CME December pound sterling futures contract
Contract size: £62,500 Expiration date: 3rd week of December Exercise price: $1.45/£ Rule for exercise: an American option
exercisable anytime until expiration
Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3e 7-6
Currency option quotationsCurrency option quotations
British pound (CME)£62,500; cents per pound
Strike Calls-Settle Puts-SettlePrice Oct Nov Dec Oct Nov Dec1430 2.38 . . . . 2.78 0.39 0.61 0.801440 1.68 1.94 2.15 0.68 0.94 1.161450 1.12 1.39 1.61 1.12 1.39 1.611460 0.69 0.95 1.17 1.69 1.94 2.161470 0.40 0.62 0.82 2.39 . . . . 2.80
Note: S0$/£ = $1.45/£
Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3e 7-7
Payoff profile of a pound callPayoff profile of a pound callat expirationat expiration
$0.24/£
$1.69/£
ST$/£
CallT$/£
In-the-money
K$/£ $1.45/£
Out-of-the-money
Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3e 7-8
Profit (loss) on a call optionProfit (loss) on a call optionat expirationat expiration
-$0.16/£
$1.45/£ $1.69/£
ST$/£
K$/£
+$0.08/£
$1.93/£ -$0.40/£
Option premiumCallt$/£ = $0.40/£
FX rate at expiration $1.45/£ $1.69/£ $1.93/£Premium (cost) -$25,000 -$25,000 -$25,000Exercise price $0 -$90,625 -$90,625Spot £ sale $0 $105,625$120,625Net profit -$25,000 -$10,000 +$5,000
Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3e 7-9
Payoff profile of a call optionPayoff profile of a call optionat expirationat expiration
Long call
ST$/£
CallT$/£
Short call
ST$/£
-CallT$/£
KT$/£
KT$/£
In-the-money
Out-of-the-
money
Out-of-the-
moneyIn-the-money
Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3e 7-10
Payoff profile of a put optionPayoff profile of a put optionat expirationat expiration
Long put
ST$/£
PutT$/£
Short put
ST$/£
-PutT$/£
KT$/£
KT$/£
In-the-money
Out-of-the-
money
Out-of-the-
money
In-the-money
Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3e 7-11
Puts and callsPuts and callsAn option to buy pounds
at KT$/£
ST$/£
CallT$/£
An option to sell dollarsat KT
£/$
ST£/$
PutT£/$
Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3e 7-12
Forwards, puts, and callsForwards, puts, and calls
ST$/£
CallT$/£
A combination of a long call and a short put at the same exercise price and with the same expiration date results in a long forward position at that forward price
FT$/£-PutT
$/£
ST$/£ST
$/£
Long call Short put Long forward
+ =
Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3e 7-13
Put-call parity:Put-call parity: Call CallTT
d/fd/f Put PutTTd/fd/f + K + Kd/fd/f = F = FTT
d/fd/f
ST$/£
CallT$/£ KT$/£-PutT
$/£
ST$/£ST
$/£
Long call Short put
Long forward
+
=
+
FT$/£
ST$/£
Exercise price
KT$/£
Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3e 7-14
The time value of an optionThe time value of an option Time value = Option value - intrinsic value
- Intrinsic value = value if exercised immediately
The time value of a currency option is a function of the following six determinants- Exchange rate underlying the option- Exercise price or striking price - Riskless rate of interest id in currency d- Riskless rate of interest if in currency f- Volatility in the underlying exchange rate- Time to expiration
Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3e 7-15
Time value and volatilityTime value and volatility
Call value
Spot rate
Intrinsic value
Time value
High volatility Low volatility
Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3e 7-16
Time value and volatilityTime value and volatility
Currency call option value
Call option value
Intrinsic value
Time value
Exchange rate
Exchange rate distribution
Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3e 7-17
The interaction of time and The interaction of time and variancevariance
If instantaneous changes are a random walk, then T-period variance is T times one-period varianceT
2 = T 2
where 2 = 1-period varianceT
2 = T-period variance Estimation of exchange rate volatility
- Historical volatility- Implied volatility
Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3e 7-18
Advanced: Pricing currency optionsAdvanced: Pricing currency options Suppose the Australian-per-US dollar
spot rate A$2.4/$ bifurcates by a continuously compounded ±4 percent per period for 4 periods
4 successive bifurcations result in 24 = 16 price paths
Value after 1 period is P1 = P0e±0.04 (A$2.4/$)e-0.04 = A$2.306/$ (A$2.4/$)e+0.04 = A$2.498/$each with 50 percent probability
Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3e 7-19
±4 percent for 4 periods±4 percent for 4 periods
+16% +12%
+8% +8%+4% +4%
0% 0%-4% -4%
-8% -8%-12%
-16%
2.8162.706
2.600 2.6002.498 2.498
2.400 2.400 2.4002.306 2.306
2.215 2.2152.129
2.045
Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3e 7-20
2244 = 16 possible price paths = 16 possible price pathsn = 1 2 3 4
11
1 41 3
1 2 61 3
1 41
12n = 2 4 8 16
Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3e 7-21
End-of-period distribution for n = 4End-of-period distribution for n = 4
0.00
0.10
0.20
0.30
0.40
2.816 2.600 2.400 2.215 2.045
Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3e 7-22
More frequent compounding…More frequent compounding…
Suppose we apply the binomial model with 1% per period for 16 periods
This results in 216 = 65,536 price paths and (n+1) = 17 possible prices
Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3e 7-23
±1 percent for 16 periods±1 percent for 16 periods
2.498 …2.473
2.448 2.448 …2.424 2.424
2.400 2.400 2.400 …2.376 2.376
2.352 2.352 …2.329
2.306 …
+5%+4%
+3%+2% +2%
+1% +1%0% 0%
-1% -1% -2% -2%
-3% -4%
Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3e 7-24
End-of-period distribution for n = 16End-of-period distribution for n = 16
0.00
0.05
0.10
0.15
0.202.
045
2.12
9
2.21
5
2.30
6
2.40
0
2.49
8
2.60
0
2.70
6
2.81
6
Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3e 7-25
The Binomial and B-S OPMsThe Binomial and B-S OPMs As the binomial process generating
up and down movements bifurcates over shorter and shorter intervals- the binomial distribution
approaches the normal distribution
- continuous-time pricing methods (e.g., the Black-Scholes OPM) can be used